Asymptotic Analysis 88 (2014) DOI /ASY IOS Press

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1 Asymptotic Analysis 88 4) 5 DOI.333/ASY-4 IOS Press Smoluchowski Kramers approximation and large deviations for infinite dimensional gradient systems Sandra Cerrai and Michael Salins Department of Mathematics, University of Maryland, College Park, MD, USA Abstract. In this paper, we explicitly calculate the quasi-potentials for the damped semilinear stochastic wave equation when the system is of gradient type. We show that in this case the infimum of the quasi-potential with respect to all possible velocities does not depend on the density of the mass and does coincide with the quasi-potential of the corresponding stochastic heat equation that one obtains from the zero mass limit. This shows in particular that the Smoluchowski Kramers approximation can be used to approximate long time behavior in the zero noise limit, such as exit time and exit place from a basin of attraction. Keywords: Smoluchowski Kramers approximation, large deviations, exit problems, gradient systems. Introduction In the present paper, we consider the following damped wave equation in a bounded regular domain O R d, perturbed by noise u ε t, ξ) = Δu ε t, ξ) u ε t, ξ) + B u ε t, ) ) ξ) + ε wq t, ξ), ξ O, t>,.) u u ε ε, ξ) = u ξ),, ξ) = v ξ), ξ O, u ε t, ξ) =, ξ O for some parameters <ε,. ere, w Q / is a cylindrical Wiener process, white in time and colored in space, with covariance operator Q,forsomeQ LL O)). Concerning the non-linearity B, we assume Bx) = Q DFx), x L O), for some F : L O) R, satisfying suitable conditions. We also consider the semi-linear heat equation { uε t, ξ) = Δu εt, ξ) + B u ε t, ) ) ξ) + ε wq t, ξ), u ε, ξ) = u ξ), * Corresponding author. cerrai@math.umd.edu. ξ O, t>, u ε t, ξ) =, ξ O..) 9-734/4/$7.5 4 IOS Press and the authors. All rights reserved

2 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations In [], it has been proved that if ε>isfixedand tends to zero, the solutions of.) converge to the solution of.), uniformly on compact intervals. More precisely, for any η>andt> lim P sup t [,T ] u ε t) u ε t) L O) >η ) =. Moreover, in the case d = andq = I, it has been proven that for any fixed ε>, the first marginal of the invariant measure of Eq..) coincides with the invariant measure of Eq..), for any >. In this paper, we are interested in comparing the small noise behavior of the two systems. More precisely, we keep > fixed and let ε tend to zero, to study some relevant quantities associated with the large deviation principle for these systems, as the quasi-potential that describes also the asymptotic behavior of the expected exit time from a domain and the corresponding exit places. Due to the gradient structure of.), as in the finite dimensional case studied in [6], we are here able to calculate explicitly the quasi-potentials V x, y) for system.) as V x, y) = Δ) / Q x L O) + F x) + Q y L O)..3) Actually, we can prove that for any > where { V x, y) = inf I z): z) = x, y), lim I z) = Q ϕ ϕ t) Δϕt) + t) B ϕt) ) ) C / zt) } =,.4) L O) with ϕt) = Π zt), and C u, v) = Δ) Q u, ) Δ) Q v, u, v) L O) O). From.4), we obtain that V x, y) Δ) / Q x L O) + F x) + Q y L O). Thus, we obtain the equality.3) by constructing a path which realizes the minimum. An immediate consequence of.3) is that for each > dt V x) = inf V x, y) = V x,)= V x),.5) y O) where V x) is the quasi-potential associated with Eq..). Now, consider an open bounded domain G L D), which is invariant for u asymptotically stable equilibrium, and for any x G let us define and is attracted to the τ ε := inf { t : u ε t) G } and τ ε = inf { t : u ε t) G }.

3 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations 3 In a forthcoming paper we plan to prove lim ε ε log Eτ ε = inf y G V y)..6) As a consequence of.5), this would imply that, in the gradient case for any fixed > it holds lim ε ε log Eτ ε = inf V y) = lim ε log Eτ ε. y G ε. Preliminaries and assumptions Let = L O) andleta be the realization of the Laplacian with Dirichlet boundary conditions in. Let{e k } k N be the complete orthonormal basis of eigenvectors of A, andlet{ α k } k N be the corresponding sequence of eigenvalues, with <α α k α k+,foranyk N. The stochastic perturbation is given by a cylindrical Wiener process w Q t, ξ), for t andξ O, which is assumed to be white in time and colored in space, in the case of space dimension d>. Formally, it is defined as the infinite sum w Q t, ξ) = + k= Qe k ξ)β k t),.) where {e k } k N is the complete orthonormal basis in L O) which diagonalizes A and {β k t)} k N is a sequence of mutually independent standard Brownian motions defined on the same complete stochastic basis Ω, F, F t, P). ypothesis. The linear operator Q is bounded in and diagonal with respect to the basis {e k } k N which diagonalizes A. Moreover, if {λ k } k N is the corresponding sequence of eigenvalues, we have k= λ k α k < +. In particular, if d = we can take Q = I, butifd> the noise has to colored in space. Concerning the non-linearity B, we assume that it has the following gradient structure. ypothesis. There exists F : R of class C, with F ) =, F x) and DF x), x for all x, such that.) Bx) = Q DF x), x. Moreover, DF x) DF y) κ x y, x, y..3) Example.. ) Assume d = andtakeq = I. Letb : R R be a decreasing Lipschitz continuous function with b) =. Then the composition operator Bx)ξ) = bxξ)), ξ O, is of gradient type. Actually,

4 4 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations if we set xξ) F x) = bη)dηdξ, x, O we have Bx) = DF x), x. Moreover, it is clear that F ) =, F x) forallx, and DF x), x = b xξ) ) xξ)dξ, x. O ) Assume now d, so that Q is a general bounded operator in, satisfying ypothesis. Let b : R R be a function of class C, with Lipschitz-continuous first derivative, such that b) = and bη), for all η R. Moreover, the only local minimum of b occurs at. Let F x) = b xξ) ) dξ, x. O It is immediate to check that F ) = andf x), for all x. Furthermore,foranyx DF x)ξ) = b xξ) ), Therefore, the nonlinearity ξ O. Bx) = Q b x ) ), x, satisfies ypothesis. For any δ R, we denote by δ the completion of C u δ = αk δ u, e k. k= This is a ilbert space, with the scalar product u, v δ = O) in the norm αk δ u, e k v, e k. k= In what follows, we shall define δ = δ δ and we shall set =. Next, for >, we define A : DA ) δ δ by A u, v) = v, Au ) v, u, v) DA ) = δ+,.4) and we denote by S t) the semigroup on generated by A.

5 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations 5 In [, Proposition.4], it has been proven that S t)isac -semigroup of negative type, namely, there exist M > andω > such that S t) Lδ ) M e ωt, t..5) Moreover, for any > we define the operator Q : δ δ by setting Q v =, Qv), v δ. Therefore, if we define ˆF u, v) = F u), ˆQu, v) = Qu, u, v), Eq..) can be rewritten as the following abstract stochastic evolution equation in the space dzε t) = [ A zε t) Q ˆQDˆF z ε t) )] dt + εq dwt), zε ) = u, v )..6) Analogously, Eq..) can be rewritten as the following abstract stochastic evolution equation in du ε t) = [ Au ε t) Q DF u ε t) )] dt + εq dwt), u ε ) = u..7) Definition.. ) A predictable process zε L Ω, C[, T ]; )) is a mild solution to.6) if z ε t) = S t)u, v ) S t s)q ˆQDˆF z ε s) ) ds + ε ) A predictable process u ε L Ω, C[, T ]; )) is a mild solution to.7) if u ε t) = e ta u Remark.3. If we define C = + e t s)a Q DF u ε s) ) ds + ε S t)q Q S t)dt, e t s)a Q dws). S t)q dws). as shown in [, Proposition 5.] we have C u, v) = A) Q u, ) A) Q v, u, v). Therefore, we get A C DˆF u, v) =, ) Q DF u) = Q ˆQDˆF u, v), u, v).

6 6 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations This means that Eq..6) can be rewritten as dzε t) = [ A zε t) + A C DˆF zε t) )] dt + εq dwt), zε ) = u, v ). In the same way, Eq..7) can be rewritten as du ε t) = [ Au ε t) + ACDF u ε t) )] dt + εq dwt), u ε ) = u, where + C = e ta QQ e ta dt = A) Q. In particular, both.6) and.7) are gradient systems for more details see [7]). As we are assuming that DF : is Lipschitz continuous, we have that Eq..6) has a unique mild solution zε L p Ω, C[, T ]; )) and Eq..7) has a unique mild solution u ε L p Ω, C[, T ]; )), for any p andt>. In [, Theorem 4.6] it has been proved that in this case the so-called Smoluchowski Kramers approximation holds. Namely, for any ε, T>andη> lim P sup u ε u) u ε t) ) >η =, t [,T ] where u ε t) = Π z ε t). 3. A characterization of the quasi-potential For any >andt <t,andforanyz C[t, t ]; ) andz,wedefine I t,t z) = inf{ ψ L t,t );) : z = z z,ψ}, 3.) where z z,ψ solves the following skeleton equation associated with.6) z z,ψ t) = S t t )z + t S t s)a C DˆF z z,ψ s)) ds + t S t s)q ψs)ds. 3.) Analogously, for t <t,andforanyϕ C[t, t ]; ) andu, wedefine I t,t ϕ) = inf{ ψ L t,t );) : ϕ = ϕ ψ,u }, 3.3) where ϕ u,ψ solves the problem ϕ u,ψt) = e t t )A u t e t s)a Q DF ϕ u,ψs) ) ds + t e t s)a Qψs)ds. 3.4)

7 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations 7 In what follows, we shall also denote I z) = sup I t, z), t< I ϕ) = sup I t, ϕ). t< Since.6) and.7) have additive noise, as a consequence of the contraction lemma we have that the family {z ε } ε> satisfies the large deviations principle in C[, T ]; ), with respect to the rate function I,T and the family {uε } ε> satisfies the large deviations principle in C[, T ]; ), with respect to the rate function I,T. In what follows, for any fixed > we shall denote by V the quasi-potential associated with system.6), namely V x, y) = inf { I,T z): z) =, zt ) = x, y), T>}. Analogously, we shall denote by V the quasi-potential associated with Eq..7), that is V x) = inf { I,T ϕ): ϕ) =, ϕt ) = x, T> }. Moreover, for any > we shall define V x) = inf V x, y) = inf { I y,t z): z) =, Π zt ) = x, T> }. O) In [3, Proposition 5.4], it has been proven that V x) can be represented as { } V x) = inf I ϕ): ϕ) = x, lim ϕt) =. ere, we want to prove a similar representations for V x, y), for any fixed >. To this purpose, we first introduce the operator L t,t : L t, t ): ),definedas L t,t ψ = t S t s)q ψs)ds. 3.5) Theorem 3.. For any >and x, y) we have { V x, y) = inf z): z) = x, y), lim C / zt) } =. 3.6) I Proof. First we observe that by the definitions of I t,t and V x, y), V x, y) = inf { I T, z): z T ) =, z) = x, y), T>}. Now, for any >andx, y),wedefine { M x, y) := inf I z): z) = x, y), lim C / zt) } =. Clearly, we want to prove that M x, y) = V x, y), for all x, y).

8 8 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations If z is a continuous path with z T ) = andz) = x, y), we can extend it in C,);), by defining zt) =, for t< T. Then, since DF ) =, we see that I z) = I T, z), so that M x, y) V x, y). Now, let us prove that the opposite inequality holds. If M x, y) =+, there is nothing else to prove. Thus, we assume that M x, y) < +. This means that for any ε>there must be some zε C,);), with the properties that zε ) = x, y) and lim C / zε t) =, I ) z ε M x, y) + ε. In what follows we shall prove that the following auxiliary result holds. Lemma 3.. For any >, there exists T > such that for any t <t T we have ImL t,t ) = DC / ) and L ) z L t,t t,t c, t );) t ) C / z, z Im L ) t,t, 3.7) where C x, y) = A) Q x, ) A) Q y. Thus, let T be the constant from Lemma 3. and let t ε < be such that C / zε t ε ) <ε. Moreover, let ψε := L t ε T,t ε ) zε t ε ). By Lemma 3. we have ψ L ε t c ε T,t ε;) ε. Next, we define ẑ ε t) = t ε T S t s)q ψ ε s)ds. Thanks to.5), we have ε t ε T ẑ ε t) ε M dt t ε T t ε T e ωt s) Qψε s) ) ds dt M ω ε Qψ ε L t t dt ε T,tε); ) ε T M T Qψ ε ω L t c ε T,tε); ) ε.

9 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations 9 Furthermore, ẑ ε t ε T ) = andẑ ε t ε ) = zt ε ). Finally, we notice that so that ẑε t) = S t s)q QDF ẑε s) ) ds + S t s)q ψ ε s) + QDF ẑε s) )) ds, t ε T t ε T I ẑ ) t ε T,t ε ε = ψ ε + QDF ẑε s) ) L t c ε T,t ε);) ε. Now if we define z ε t) = { ẑ ε t) ift ε T t<t ε, zε t) ift ε t, we see that z ε Ct ε T,);) and V x, y) I ) t ε T, z ε = I ẑ ) t ε T,t ε ε + I ) t ε, z ε c ε + M x, y) + ε. Due to the arbitrariness of ε>, we can conclude. Proof of Lemma 3.. It is immediate to check that L t,t ) z L t,t );) = therefore, since Q u, v) = A) Qv, t Q Ss)z ds u, v) see [, Section 5] for a proof), we can conclude L ) z t,t L t,t = t );) Q A) Π Ss)z ds. Now, if we expand S t) into Fourier coefficients, by [, Proposition.3], we get S u, v) = ˆf k t)e k, ĝ k t)e k), k= where d ˆf k dt t) = ĝ k t), ˆf k ) = u k = u, e k, dĝ k dt t) = α k ˆf k t) ĝ k t), ĝ k ) = v k = v, e k.

10 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations From these equations we see that ĝ k t) = α k d ˆf k dt t) d ĝ k dt t). This means that L ) z t,t L t,t );) λ k = k= u k + λ k α k αk v k λ k ˆf k α t t ) λ k k αk ĝ k t t ) ) = Notice that C / z C / S t t )z ). 3.8) ) C / z / C z + ) C / z and that C / commutes with S t). Therefore, by.5) C / St)z S t)c / z + M e ωt C / z. According to 3.8), this implies L ) z t,t L t,t );) so that, if we take T > suchthat ) + M e ωt < ) + ) M e ωt t ) C / z, we can conclude. 4. The main result If z C,];) is such that I, z) < +, thenwehave I z) = Q ϕ ϕ t) + t) Aϕt) + Q DF ϕt) ) ) dt. 4.)

11 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations Actually, if I, z) < +, then there exists ψ L,);) such that ϕ = Π z is a weak solution to ϕ ϕ t) = Aϕt) t) Q DF ϕt) ) + Qψ. This means that ψt) = Q ϕ ϕ t) + t) Aϕt) + Q DF ϕt) ) ) and 4.) follows. By the same argument, if I, ϕ) < +, then it follows that I ϕ) = ϕ Q t) Aϕt) + Q DF ϕt) ) ) dt. 4.) Theorem 4.. For any fixed > and x, y) D A) / Q ) DQ ) it holds V x, y) = A) / Q x + F x) + Q y. 4.3) Moreover, V x) = A) / Q x + F x). 4.4) In particular, for any >, V x) := inf V x, y) = V x,)= V x). y Proof. First, we observe that if ϕt) = Π zt), then I z) = + Q ϕ ϕ t) t) Aϕt) + Q DF ϕt) ) ) dt ) ϕ Q t), Q ϕ t) Aϕt) + QDF ϕt) ) dt. 4.5) Now, if then lim C / lim zt) =, A) / Q ϕt) + Q ϕ t) =,

12 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations so that This yields Q ϕ t), Q ) ϕ t) Aϕt) = A) / Q ϕ) + F ϕ) ) + + QDF ϕt) ) dt ). Q ϕ V x, y) A) / Q x + F x) + Q y. Now, let zt) be a mild solution of the problem zt) = S t)x, y) S t s)q QDF zs) ) ds, and let x, y) DC / ). Then ϕt) = Π zt) is a weak solution of the problem ϕ ϕ t) = A ϕt) t) Q DF ϕt) ), ϕ) = x, Moreover, as proven in Lemma 4., lim C / zt) =. Then, if we define ˆϕt) = ϕ t)fort, we see that ˆϕt)solves ˆϕ ˆϕ t) = A ˆϕt) + t) Q DF ˆϕt) ), ˆϕ) = x, Thanks to 4.5) this yields and then I ˆϕ) = A) / Q x + F x) + Q y ϕ ) = y. ˆϕ ) = y. V x, y) = A) / Q x + F x) + Q y. As known, an analogous result holds for V x). In what follows, for completeness, we give a proof. We have I ϕ) = ϕ Q t) + Aϕt) Q DF ϕt) ) ) dt ϕ + Q t), Q Aϕt) + Q DF ϕt) )) dt. 4.6)

13 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations 3 From this we see that V x) A) / Q x + F x). Just as for the wave equation, for x D A) / Q ), we define ϕ to be the solution of ϕt) = e ta x e t s)a Q DF ϕs) ) ds. We have lim t + A) / Q ϕt) =. Then, if we define ˆϕt) = ϕ t)weget so that and ˆϕ t) = A ˆϕt) + Q DF ˆϕt) ), I ˆϕ) = A) / Q x + F x) V x) = A) / Q x + F x). Now, in order to conclude the proof of Theorem 4., we have to prove the following result. Lemma 4.. Let x, y) D A) / Q ) DQ ) and let ϕ solve the problem ϕ ϕ t) = Aϕt) t) Q DF ϕt) ), ϕ) = x, Then zt) = ϕt), ϕ ) t) D A) /) D Q ), t ϕ ) = y. 4.7) and lim C / t + zt) =. 4.8) Proof. If in 4.7) we take the inner product with Q ϕ/t), we have Q ϕ t) = d Q ϕt) dt + A) / Q ϕt) + F ϕt) )). 4.9)

14 4 S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations Therefore, if we define Φ x, y) = A) / Q x + Q y + F x), as a consequence of 4.9) we get Φ zt) ) Φ u, v). 4.) Next, by 4.7) and the assumption that DFx), x, we calculate that d dt Q ϕ ) t) + ϕt) = Q ϕ ) t) + ϕt), Q Aϕt) QDF ϕt) ) Q A) / ϕt) d Q A) / ϕt) dt d dt F ϕt) ). A consequence of this is that Q A) / ϕt) dt Q y + Q x + Q A) / x + F x). 4.) Now, if zt) = ϕt), ϕ t)), for any t, T>wehave zt + t) = S t)zt ) By.5) and.3) we have T +t C / T T +t c c T T +t T T +t T T +t T S t + T s)q QDF ϕs) ) ds S T + t s)q QDF ϕs) ) ds. S t + T s)q A) / DF ϕs) ) ds e ωt+t s) A) / QDF ϕs) ) ds e ωt+t s) ϕs) ds c ϕ L T,T +t);). Therefore, by 4.), for any ε> we can pick T ε > large enough so that for all t> C / Tε+t T ε S t + T ε s)q QDF ϕs) ) ds < ε.

15 Next, by.5), Then, as S. Cerrai and M. Salins / Smoluchowski Kramers approximation and large deviations 5 C / S t)zt ) M e ωt C / zt ). C / z cφ z), z, by 4.) we can find a t ε large enough so that for all T>andt>t ε C / S t)zt ) < ε. Then for t>t ε + t ε C / zt) <ε whichiswhatweweretryingtoprove. References [] S. Cerrai and M.I. Freidlin, On the Smoluchowski Kramers approximation for a system with an infinite number of degrees of freedom, Probability Theory and Related Fields 35 6), [] S. Cerrai and M.I. Freidlin, Smoluchowski Kramers approximation for a general class of SPDE s, Journal of Evolution Equations 6 6), [3] S. Cerrai and M. Röckner, Large deviations for invariant measures of stochastic reaction diffusion systems with multiplicative noise and non-lipschitz reaction term, Annales de l Institut enri Poincaré Probabilités et Statistiques) 4 5), [4] Z. Chen and M. Freidlin, Smoluchowski Kramers approximation and exit problems, Stoch. Dyn. 5 5), [5] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 99. [6] M. Freidlin, Some remarks on the Smoluchowski Kramers approximation, J. Statist. Phys. 7 4), [7] M. Fuhrman, Analyticity of transition semigroups and closability of bilinear forms in ilbert spaces, Studia Mathematica 5 995), 53 7.